Probability
Event and Algebra of Events
Grade 12
Question:
<p>If <i>A</i> and <i>B</i> are two mutually exclusive events, then</p>
<p>(1) \(P(A) \leq P(\bar{B})\)</p>
<p>(2) \(P(A) > P(B)\)</p>
<p>(3) \(P(B) \leq P(\bar{A})\)</p>
<p>(4) \(P(A) > P(B)\)</p>
Step-by-Step Solution
Key Concept: Mutually exclusive events cannot occur simultaneously, so P(A ∩ B) = 0, which means P(A ∪ B) = P(A) + P(B). This is the fundamental property that distinguishes mutually exclusive events from independent events.
<p><strong>Step 1:</strong> For mutually exclusive events A and B, by definition: A ∩ B = ∅ (empty set)</p><p><strong>Step 2:</strong> Therefore: P(A ∩ B) = 0</p><p><strong>Step 3:</strong> Using addition rule: P(A ∪ B) = P(A) + P(B) - P(A ∩ B) = P(A) + P(B) - 0 = P(A) + P(B)</p><p><strong>Step 4:</strong> Also, P(A) + P(B) ≤ 1 (since both are subsets of sample space)</p><p><strong>Correct Statements:</strong></p><p><strong>[1]</strong> P(A ∩ B) = 0 ✓ (Definition of mutually exclusive)</p><p><strong>[2]</strong> P(A ∩ B) = P(A)·P(B) ✗ (This is independence, not mutual exclusivity)</p><p><strong>[3]</strong> P(A ∪ B) = P(A) + P(B) ✓ (Direct consequence of mutual exclusivity)</p><p><strong>[4]</strong> P(A|B) = P(A) ✗ (Would imply independence; actually P(A|B) = 0 since A cannot occur if B occurs)</p><p>∴ Answer: 1, 3</p>
Correct Answer: 1,3